Caliber of space of subtle complete coupled systems (Q2062116)

A coupled system on a topological space \(X\) is a collection of closed subsets such that the intersection of any two is non-empty. The collection of all maximal coupled systems, denoted \(\lambda X\), may be given a natural topology. A maximum coupled system is subtle provided it contains at least one finite subset, and the subspace of these is denoted by \(\lambda^*X\). When \(X\) is compact, say that a coupled system \(\xi\) is complete provided for any closed \(C\subset X\), \(C\in \xi\) whenever every neighbourhood of \(C\) contains a member of \(\xi\): this gives rise to the space \(NX\) of complete coupled systems and thence to the subspace \(N^*X\) of subtle ones. For an infinite compact space \(X\) it is shown that the calibres of \(\lambda^*X\) and \(N^*X\) are the same. The weight and Souslin number of a related space \(N^d_{\aleph_0}X\) are also considered and shown to agree with the corresponding cardinalities of \(X\).